Probability/Chance up if applying more Ivies schools?

Parents Forum
41

<p>It is NOT the same as the gambler’s fallacy.</p>

<p>The reason why applying to multiple Ivies would (potentially) increase your chances, would be that you don’t know for sure which one is looking for a candidate like you. What I’m saying is, you may be in a situation where you have an 80% chance of being admitted to some of the Ivies, but only a 10% chance of being admitted at some of the others, and you would never know which one you had the greater chance of admittance to, not knowing the rest of the applicant pool/the admission counselors’ frame of minds/etc. But if you applied to both Harvard and Dartmouth, rather than just Dartmouth or just Harvard, your chances of applying to the one that wants YOU (assuming one does want you) i.e. the one for which your chance of acceptance is much higher, will go up. </p>

<p>So, applying to multiple ones, would increase your chance of finding the one (assuming there WAS one) that wants you, because there is NOT an equal chance of you being accepted to each one. </p>

<p>Say you are an athlete.</p>

<p>If Dartmouth is looking for a sporty kid, your chance of being accepted there may be 80%.</p>

<p>If Harvard is looking for an artsy kid, your chance of being accepted would be 10%. </p>

<p>Etc.</p>

<p>This is very simplified but the idea is that you have nowhere NEAR an equal chance of getting admitted to each Ivy, assuming your stats are acceptable, nor nowhere near the “avg. admit rate”. Your admission possibility is distinct and individualized at each school, and you can’t know which one you have the best chance at. So apply to all of them, and if there is one you have a good chance at, you’ll hit it.</p>

42

<p>Quant, you admit that pH, pY and pP are not independent. What possible justification do you have for multiplying the three probabilities to get their joint probability? That only works when the events are independent. These aren’t.</p>

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<p>Yes, but the a priori value of pH <em>is</em> affected by the value of pY, because students who are more attractive to Harvard are also more attractive to Yale, in general.</p>

43

<p>There are at least two distinct intuitive concepts of the probability of admission to HYP, which lead to different ideas of “independence” of the decisions.</p>

<p>First, let me take the case that I think CardinalFang and (in part) tokenadult are considering. Suppose (just for the sake of an illustrative analysis) that 20,000 students apply to both Harvard and Yale. Of these students, Harvard takes 1250 and Yale takes 1250, but there is only a partial overlap between Harvard’s 1250 and Yale’s. (The rest of the admitted students at each college did not apply to the other.)</p>

<p>If we look at the decisions, we could set them up in a table:</p>

<p>Outcomes-----Yale Yes—Yale No
Harvard Yes----1000--------250
Harvard No------250-----18,500 </p>

<p>That is, 1000 students were admitted to both Harvard and Yale, 250 to Yale but not to Harvard, 250 to Harvard but not to Yale, and 18,500 to neither (for the sake of the analysis). Under this scenario, if a student is in the “Harvard Yes” row, then 1000 times out of 1250, the student is also admitted to Yale, giving a probability of 4/5. If the student is in the “Harvard No” row, then 18,500 times out of 18,750 times, the student is also in the “Yale No” column. This means that a student declined at Harvard has only a 1/75 chance of admission to Yale–if I did the arithmetic right. (I’ve set it up to be symmetric in the two colleges.) Then we would not say that the outcomes are statistically independent. </p>

<p>However, the 4/5 and the 1/75 are not the pY values that I was writing about. I will explain that aspect more fully below.</p>

44

<p>DS asked me this question and my answered was “remotely or 7-10%” as normal accepted rate and he argued with me I should use accepted rate of (1 - (1 -pH)(1 - pY)(1 - pP)). So this was how started. In this case should be around 24%. pH - 7.9%, pY - 9.9%, pP - 8.6% from US News. Any input for help to defend me or I am too old to argue with him.</p>

45

<p>If you’re not up to par for an Ivy League school, applying to all eight of them won’t help. If you are a competitive applicant, then applying to a lot of them will help negate the crapshoot factor, and your chances are probably better.</p>

46

<p>In the example above, if we know that a student has been admitted to Harvard, from the table, we would say that the student has a 4/5 chance of being admitted to Yale. Why is this not the same as pY in my earlier comments?</p>

<p>The 4/5 chance is identical for all students who were admitted by Harvard, regardless of their specific personal qualifications. In contrast, the pY value is supposed to represent the true, underlying probability of admission to Yale for each particular student, taking all qualifications (and the qualifications of other applicants) into account. We don’t know the pH values of the students admitted to Harvard, though we can surmise that in general they were substantially higher than the pH values of students not admitted (though inversions could occur–i.e., “yes” to an applicant whose personal, individual probability is lower than an applicant who is rejected). </p>

<p>The probability pY for a particular applicant reflects the applicant’s qualifications and the quality of the application itself. However, I treat this pY as a probability only, due to the inherent variability of the admissions process. Normally, a probability is determined empirically by repeating an experiment a very large number of times, counting the times a particular outcome is observed, and then taking the ratio of the number of times the outcome is observed to the number of times the experiment has been run (actually, in the limit of an infinite number of times of running the experiment). Clearly, the values of pY and pH cannot be determined empirically for an applicant, because the admissions committee members would remember him/her after the first time the application was read (or after the nth time, anyway).</p>

<p>Some might deny that pY and pH exist, on the grounds that they cannot be determined by experiment. That is a valid viewpoint, which I respect.</p>

<p>However, it is my view that some students would be admitted to Harvard no matter when their files were read, no matter by whom they were read, no matter what other students with similar interests were applying . . . Some students would never be admitted, regardless. But for some students, I do think that if “replicate” applications could be evaluated, the outcomes would vary. Hence, I see this as inherently probabilistic, and pY and pH are intended to reflect that.</p>

<p>In this sense, pY and pH are determined by the quality of the application (and of others’ applications) going into the selection process, before any decisions have been reached. They are set values, for any particular applicant, once the application and recommendations have been submitted for all applicants. Yes, pH and pY are correlated with each other. </p>

<p>What I am claiming is that pY is not changed by the outcome of the Harvard decision. For example, suppose that an applicant has a 0.99 probability of being admitted to Harvard, pH = 0.99. Then pY is probably similarly high. But suppose that the random outcome with a probability of only 1% does occur, and the student is declined at Harvard. Does pY drop for that specific applicant? I’d say no. That student will most probably wind up contributing to the number in the “Yale yes, Harvard no” box.</p>

47

<p>soccer92boy, the difficulty is that the raw odds of admission are not the same as the true probabilities, for the overwhelming majority of applicants. What would be best to compare are the odds of admission if applying to a single highly selective college, vs. two, vs. three . . ., with realistically chosen values of the probabilities–though at some point, the application-fatigue-Heisenberg-uncertainty principle noted by vicariousparent begins to apply . . . and more so, if the GC is annoyed by the large number of applications.</p>

48

<p>What makes a student have an a priori probability of, say, 20% of being admitted to Harvard? Aren’t quite a few of the factors going to be the same for her Yale application? Her main essay, which either was one of her best essays or not? Her recommenders, who either praised her to the skies or were just that little bit lukewarm? Her guidance counselor’s recommendation, which goes both to Harvard and to Yale? The quality of the other applicants, again probably roughly the same for Harvard and Yale? </p>

<p>It’s the same application going to both schools. That’s why the a priori probabilities aren’t independent.</p>

49

<p>I agree that pH and pY are not independent. (I’ve noted several times that they are correlated.) Also, viewed in terms of the admit table, the outcomes are not statistically independent–which is what tokenadult was driving at, I think. </p>

<p>However, I am claiming that each applicant has a personal, inherent probability to be admitted to each school, and that the decisions at the various schools are made independently of each other–even though a particular applicant may have all high values of pH, pY, and pP, or all low values. Then the overall probability that I calculated is correct.</p>

<p>It is possible for probabilities to be correlated, yet for the results of successive trials to be independent of each other. If a fair coin is tossed twice, the probability for it to come up heads the first time is 0.50 and the probability for it to come up heads the second time is 0.50. These two probabilities not independent–in fact, they are identical. Yet the outcomes of the two tosses, conditioned on these probabilities, are independent.</p>

<p>I think the continuing discussion reflects the difficulties of writing precisely about mathematics, in English.</p>

50

<p>Yes, I think what QuantMech wrote in [post</a> # 39](<a href=“http://talk.collegeconfidential.com/1064072406-post39.html]post”>http://talk.collegeconfidential.com/1064072406-post39.html) correctly reads what I was trying to say, and leads into several further posts from his keyboard that draw distinctions that are sometimes missed in these discussions. It is hard to write about this issue for a general audience, when the word “independent” has both a colloquial meaning and a technical meaning. </p>

<p>I appreciate what vicariousparent writes about application going down as application number goes up. Apply to a college you would really like to attend, and apply well to any college to which you submit an application at all, but don’t spread yourself too thin.</p>

51

<p>I agree with a lot of what QuantMech has said on this thread.</p>

<p>As an aside, I think someone somewhere does have actual data on the number of cross-admits between H and Y. If not publicly available, then someone could look at the results threads for H and Y and come up with an estimate for the C.C. population. This might help plug in actual numbers into the table in post#43. </p>

<p>The number of students applying to both schools may be very hard to come by, though one might be able to make some educated guesses.</p>

52

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<p>A gambler knows perfectly well that buying 10 lottery tickets is more likely to result in winning the big prize than just buying 1 ticket.</p>

<p>Anyway, think of it another way. Your chance of admission is zero if you don’t apply. But even the most mediocre candidate has a non-zero chance of admission if they do apply (even if just through error or perversity on part of the admission people).</p>

53

<p>OK, I see where we disagree, QuantMech. You are imagining the experiment of Applicant A, having already filled out a complete application, sending that application over and over again (as it were) to Harvard and Yale. You correctly say that in this repeated experiment, Harvard would accept the applicant pH of the time, and Yale would accept the applicant pY of the time, and those are uncorrelated. You are looking at probabilities once the applications hit the adcoms desks at Harvard and Yale.</p>

<p>I am imagining a slightly different experiment. I think of Applicant A <em>before she has filled out her application</em>. That is when I’m assigning probabilities. And they will be correlated, because, as part of the experiment, she fills out the application. Included in both universities’ probabilities will be the chance that she got a bad recommendation from the guidance counselor or wrote a bad essay. In my experiment, both universities will be more likely to accept her if her essay-- which she hasn’t written yet-- is good. So, for example, suppose there is a 90% probability that she will write a fabulous essay, and a 10% probability that she will get riproaring drunk and write the worst essay in the world and both universities will drop her application in the trash. And then there is also an individual random factor for each university, as you cite-- maybe they are more likely to accept her if they happen to read her application in the morning, or if her application happens to be read by someone who likes hikers and she’s a hiker, or some other random reason. But still, in my experiment, the probabilities are correlated. And in yours, they are not.</p>

54

<p>^^Cardinal Fang, did you miss this:</p>

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55

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Then wouldn’t it make a lot more sense to do research to find out what the colleges are likely looking for, than to waste time and effort on applications to the colleges that clearly are NOT looking for those qualities?</p>

<p>My problem with this “chance” analysis is that that the students who are most likely to be accepted are those who look at the application process strategically. </p>

<p>If we are going to stick with gambling analogies, you might ask which gambler has a better “chance” of winning at the blackjack table – the one who views it as a game of chance, and assumes that odds of winning go up with repeated deals? Or the one who views it as a game of strategy, studies the game, and pays close attention to what is being done by the other players at the table as well as the cards he is dealt over successive rounds?</p>

<p>The Ivies are NOT the same – each has its own set of institutional priorities. The most logical thing for a student to do to gain admission would be to accomplish something during high school that is significant and will set the student apart from peers, and then target the college applications to colleges that are likely to value that something.</p>

56

<p>I was always terrible at (and didn’t like) math so I don’t understand the statistics theory but I do believe in the fishbowl theory. Unbelievably talented and impressive applicants get into Ivies because they are unbelievably talented and impressive. They make up maybe 25% of the Ivy students. The other 75% are just good all around exceptional students and their names are put in fishbowl at each school and are more or less pulled randomly.</p>

<p>My son applied to all Ivies plus Stanford and MIT. His name wasn’t pulled from any of those fishbowls except Harvard and Cornell - rejected by all others. He’s now a junior at Harvard majoring in math (how ironic, huh).</p>

<p>My fishbowl theory is sort of supported by the YouTube/On Harvard Time video interview with the Harvard dean of admissions. He calls them “good all arounders”.</p>

<p>Lesson - If you want to get admitted to one, apply to them all so you are in as many fishbowls as possible.</p>

57

<p>Sorry to drag this out, but I thought of another analogy that might help to explain why the probabilities might be correlated with each other, but the individual probabilistic events (the admissions decisions at each of the schools) are independent.</p>

<p>Suppose someone is drawing one card from deck of playing cards #1, and one card from deck of cards #2 (jokers removed in both cases), and further suppose that the desirable outcome is drawing a red card.</p>

<p>As the game is set up, player A has lower than normal odds of success = drawing a red card. In deck #1 for player A, all of the hearts have been removed. Player A’s odds of success are 1/3, with that deck. In deck #2 for player A, all of the hearts except the ace have been removed. Player A’s odds with that deck are 7/20. Player A draws from the two decks. Whether a red card or a black card is actually drawn from deck #1 does not influence the outcome of the draw from deck #2.</p>

<p>Player B, on the other hand, has higher than typical odds of success. All of the clubs have been removed from Player B’s deck #1. The odds that Player B will draw a red card are 2/3. In deck #2, all of the clubs except for the ace have been removed. The odds that Player B will draw a red card from deck #2 are 13/20.</p>

<p>If you repeat this scenario many times, with slightly different odds for the players, but with the odds correlated between decks #1 and #2 for each individual player, and then set up the game matrix:</p>

<p>Red card?-------Deck #1 Yes--------Deck #1 No
Deck #2 Yes---------a-------------------b
Deck #2 No----------c-------------------d</p>

<p>where a, b, c, and d are the numbers of individuals in each category, then you will find that the outcomes are <em>not</em> statistically independent of each other. This is like the observation that the outcomes of the college admissions decisions are not statistically independent, because the applications from different applicants differ in quality (and there could be some deck-to-deck or school-to-school variations as well).</p>

<p>However, once the decks have been handed to the player (all apps are in, from everyone), then the outcome of the draw from deck #1 does not influence the outcome of the draw from deck #2.</p>

<p>If someone offers a player a third draw from another deck, are they more likely to draw a red card than they personally would be, if they had only two decks? Yes, unless their decks are stacked all black or all red.</p>

<p>In the “sorry, sorry” category, I have a few more comments on probabilities, but will reserve those.</p>

58

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<p>Yes - limit of 10 schools which should be enough for 2-3 reaches and safeties plus 4-6 matches. In some special cases they allow more - ie. if the kid is applying in several countries (this is an International school).<br>
I thought this was fairly common.</p>

<p>My two kids only applied to 8 and 7</p>

59

<p>I’m glad to see this thread, as I’ve wondered about this topic for a long time.</p>

<p>My sense is that chance plays a much larger role in selective admissions than most in CC community would care to admit, especially as the number of applications to some of these schools climbs to over 30,000. Too many stories (like @onedown&onetogo), where applicant is admitted to one or two of several highly selective schools.</p>

<p>When people (generally people who haven’t been through the process recently) wonder why my child applied to so many schools (11), I have to ask, why not? The cost of another application seems so small compared to what’s at stake.</p>

60

<p>QuantMech, you argue that at some point in time in the admissions process, the two schools are each making their decision independently, so at that time their decision probabilities are independent. I see your argument, and now we do not disagree on any math, but consider that in reality, no such point in time exists.</p>

<p>The candidate submits her two applications. Now the decisions are out of her hands, right? No, they are not. The colleges are still getting information on her. If she works hard at her sport (for which she was not recruited), unexpectedly gets selected as a replacement to the US Olympic team, wins a gold medal and becomes America’s sweetheart, both colleges find out in time to take that into consideration. If she fails a math test, gets furious at her math teacher and shoots him and eleven classmates, both colleges find out. If instead of working on her applications for her eighteenth through twenty-third colleges, she works hard and wins an academic competition, she will certainly update her applications at both schools.</p>